Alessandro Scagliotti

Cristina Cipriani, Massimo Fornasier, Alessandro Scagliotti

The connection between Residual Neural Networks (ResNets) and continuous-time control systems (known as NeurODEs) has led to a mathematical analysis of neural networks which has provided interesting results of both theoretical and practical significance. However, by construction, NeurODEs have been limited to describing constant-width layers, making them unsuitable for modeling deep learning architectures with layers of variable width. In this paper, we propose a continuous-time Autoencoder, which we call AutoencODE, based on a modification of the controlled field that drives the dynamics. This adaptation enables the extension of the mean-field control framework originally devised for conventional NeurODEs. In this setting, we tackle the case of low Tikhonov regularization, resulting in potentially non-convex cost landscapes. While the global results obtained for high Tikhonov regularization may not hold globally, we show that many of them can be recovered in regions where the loss function is locally convex. Inspired by our theoretical findings, we develop a training method tailored to this specific type of Autoencoders with residual connections, and we validate our approach through numerical experiments conducted on various examples.

Kexin Jin, Jonas Latz, Chenguang Liu et al.

The training of modern machine learning models often consists in solving high-dimensional non-convex optimisation problems that are subject to large-scale data. In this context, momentum-based stochastic optimisation algorithms have become particularly widespread. The stochasticity arises from data subsampling which reduces computational cost. Both, momentum and stochasticity help the algorithm to converge globally. In this work, we propose and analyse a continuous-time model for stochastic gradient descent with momentum. This model is a piecewise-deterministic Markov process that represents the optimiser by an underdamped dynamical system and the data subsampling through a stochastic switching. We investigate longtime limits, the subsampling-to-no-subsampling limit, and the momentum-to-no-momentum limit. We are particularly interested in the case of reducing the momentum over time. Under convexity assumptions, we show convergence of our dynamical system to the global minimiser when reducing momentum over time and letting the subsampling rate go to infinity. We then propose a stable, symplectic discretisation scheme to construct an algorithm from our continuous-time dynamical system. In experiments, we study our scheme in convex and non-convex test problems. Additionally, we train a convolutional neural network in an image classification problem. Our algorithm {attains} competitive results compared to stochastic gradient descent with momentum.

Alessandro Scagliotti, Sara Farinelli

In this paper, we consider the problem of recovering the $W_2$-optimal transport map T between absolutely continuous measures $μ,ν\in\mathcal{P}(\mathbb{R}^n)$ as the flow of a linear-control neural ODE, where the control depends only on the time variable and takes values in a finite-dimensional space. We first show that, under suitable assumptions on $μ,ν$ and on the controlled vector fields governing the neural ODE, the optimal transport map is contained in the $C^0_c$-closure of the flows generated by the system. Then, we tackle the problem under the assumption that only discrete approximations of $μ_N,ν_N$ of the original measures $μ,ν$ are available: we formulate approximated optimal control problems, and we show that their solutions give flows that approximate the original optimal transport map $T$. In the framework of generative models, the approximating flow constructed here can be seen as a `Normalizing Flow', which usually refers to the task of providing invertible transport maps between probability measures by means of deep neural networks. We propose an iterative numerical scheme based on the Pontryagin Maximum Principle for the resolution of the optimal control problem, resulting in a method for the practical computation of the approximated optimal transport map, and we test it on a two-dimensional example.

Alessandro Scagliotti, Federico Scagliotti, Laura Deborah Locati et al.

In this paper, we explore the application of ensemble optimal control to derive enhanced strategies for pharmacological cancer treatment, and we tackle the problem of the long-term management of the disease, i.e., when the complete eradication of the tumor is not achievable. In particular, we focus on moving beyond the classical clinical approach of giving the patient the maximal tolerated drug dose (MTD), which does not properly exploit the fight among sensitive and resistant cells for the available resources. Here, we employ a Lotka-Volterra model to describe the competing subpopulations, and we enclose this system within the ensemble control framework. In the first part, we establish general results suitable for application to various cancers. Then, we carry out numerical simulations in the setting of prostate cancer treated with androgen deprivation therapy, yielding a computed policy that is reminiscent of the medical `active surveillance' paradigm. Finally, inspired by the numerical evidence, we propose a variant of the celebrated adaptive therapy (AT), which we call `Off-On' AT.

Cristina Cipriani, Alessandro Scagliotti, Tobias Wöhrer

In this paper, we address the adversarial training of neural ODEs from a robust control perspective. This is an alternative to the classical training via empirical risk minimization, and it is widely used to enforce reliable outcomes for input perturbations. Neural ODEs allow the interpretation of deep neural networks as discretizations of control systems, unlocking powerful tools from control theory for the development and the understanding of machine learning. In this specific case, we formulate the adversarial training with perturbed data as a minimax optimal control problem, for which we derive first order optimality conditions in the form of Pontryagin's Maximum Principle. We provide a novel interpretation of robust training leading to an alternative weighted technique, which we test on a low-dimensional classification task.

Alessandro Scagliotti, Thomas M. Surowiec

A number of important modern applications in optimal control can be formulated as open loop control problems in which the underlying dynamical systems are subject to random inputs. These so-called ensemble control problems require the corresponding optimal control to be deterministic, as it must be computed before the realization of uncertainty and the passage of time. Practical applications of ensemble control include quantum control and the training of Neural ODEs. However, the standard approach to ensemble control treats the uncertainty in the objective function via the expectation, which provides optimal controls that only work well on average while ignoring critical outlier phenomena. This study provides a comprehensive mathematical treatment of risk-averse ensemble control. Within this setting, we adopt a control-affine structure that ensures the lower semi-continuity needed for proving the existence of optimal solutions. The central analytical contribution of this paper is a rigorous characterization of the control-to-state mapping in which we establish weak-to-strong continuity, continuous Fréchet differentiability, and weak-to-strong continuity of the derivative operator. Furthermore, this regularity yields primal and dual first-order optimality conditions characterized by an adjoint state of bounded variation, and it fulfills the functional prerequisites required for the convergence of infinite dimensional optimization algorithms. We conclude by validating these theoretical developments through a numerical experiment in quantum control.

Alessandro Scagliotti

In this paper we propose a Deep Learning architecture to approximate diffeomorphisms diffeotopic to the identity. We consider a control system of the form $\dot x = \sum_{i=1}^lF_i(x)u_i$, with linear dependence in the controls, and we use the corresponding flow to approximate the action of a diffeomorphism on a compact ensemble of points. Despite the simplicity of the control system, it has been recently shown that a Universal Approximation Property holds. The problem of minimizing the sum of the training error and of a regularizing term induces a gradient flow in the space of admissible controls. A possible training procedure for the discrete-time neural network consists in projecting the gradient flow onto a finite-dimensional subspace of the admissible controls. An alternative approach relies on an iterative method based on Pontryagin Maximum Principle for the numerical resolution of Optimal Control problems. Here the maximization of the Hamiltonian can be carried out with an extremely low computational effort, owing to the linear dependence of the system in the control variables.